Magnetic resonance reversals in optically pumped alkali-metal vapor

F. Gong, Y.-Y. Jau, and W. HapperDepartment of Physics, Princeton University, Princeton, New Jersey 08544, USA

�Received 30 January 2007; published 25 May 2007�

We report an unusual phenomenon, peculiar sign reversals of the ground-state magnetic resonances and ofthe zero-dip resonance �Zeeman resonance at zero field� of optically pumped, alkali-metal vapors. Theseanomalies occur when a weak circularly polarized D1 laser light is tuned to pump atoms predominantly fromthe lower ground-state hyperfine multiplet. One can understand the signal reversals in a simple, semiquantita-tive way with reference to the spin-temperature distribution. Quantitative computer simulations are in excellentagreement with observations.

DOI: 10.1103/PhysRevA.75.053415 PACS number�s�: 32.80.Bx, 32.30.Dx, 32.10.Fn, 32.30.Bv

In this paper, we describe an unusual phenomenon that wehave noticed for alkali-metal vapor that is optically pumpedwith D1, circularly polarized laser light. For certain opticalfrequencies, sufficiently weak light causes the vapor to be-come more opaque, while more intense light of the samefrequency causes the vapor to become more transparent. Thisis in marked contrast to all previous observations that weknow of, where pumping with circularly polarized D1 lightincreases the transparency of the vapor.

In Fig. 1 we review how circularly polarized resonancelight would pump a hypothetical alkali-metal atom with nonuclear spin. Circularly polarized light “burns out” the spin-down atoms of the ground state. When the optical pumpinglight greatly exceeds the spin relaxation rate of ground-stateatoms, nearly all the atoms will be pumped to the spin-upstate. This decreases the opacity of the atoms for D1 lightand increases the opacity for D2 light.

It is convenient to define a photon absorption cross sec-tion �=�op��� for photons of frequency � such that the pho-ton absorption rate per unpolarized atom is

�op = �opS/h� . �1�

Here S is the optical energy flux �erg cm−2 s−1�. Because theoptical pumping can spin-polarize the atom, the actual ab-sorption rate �� ��, where � � is the photon absorption op-erator �1�, can be larger or smaller than �op, and we write itas

�� �� = �A��op. �2�

The specific absorption A�� � /�op is a ground-state spinoperator with an expectation value �A�=1 for unpolarizedatoms. If a hypothetical alkali-metal atom with no nuclearspin is pumped with �+ light, the specific absorption operatorfor D1 pumping light is A=1−2Sz, where Sz is the longitu-dinal electron spin operator of the ground-state atoms; forD2 light, A=1+Sz. In Fig. 1 we have assumed that the re-population pumping comes from spontaneous decay withnegligible collisional depolarization or quenching of the ex-cited atoms. At high buffer gas pressures, where the repopu-lation pumping by D2 light is suppressed, Fricke et al. �2�have shown that the sign of the spin polarization reverses.We assume that diffusion to the walls and collisions withbuffer gas atoms cause the longitudinal spin to relax at the

rate �Sz�=−�s�Sz�. Then the specific absorption for D1 lightis �A�=3�s / �2�op+3�s�, while for D2 light we find �A�= �3�op+6�s� / �2�op+6�s�. As indicated in Fig. 1, increasingthe intensity of the circularly polarized pumping light causesthe specific absorptions to approach the limits �A�→0 for D1light and �A�=3/2 for D2 light. In both cases the limit isapproached monotonically.

With this ideal behavior in mind, we turn to what is actu-ally observed experimentally for real alkali-metal atoms withI�0. Our apparatus is shown in Fig. 2. A Pyrex glass cellcontaining a small amount of Cs metal and buffer gas�50 torr N2 and 280 torr Ar� was used. The space betweenthe front and rear walls of the cell is l=0.1 cm. The cell wasmounted in a temperature-controlled, air-heated nonmagneticoven. Three sets of Helmholtz coils were used to cancel theambient magnetic field �down to no more than 20 mG� andto produce a small longitudinal static field B, parallel to thepumping light. To drive magnetic resonances with �f = ±1,microwaves from a frequency synthesizer were beamed ontothe cell from a horn antenna located 10 cm away. The lightfrom a diode laser, stabilized by an external cavity, and op-erating at the 895-nm Cs D1 line, was circularly polarized,and neutral density filters were used to control the beamintensity. The light transmitted through the optically pumpedCs cell was continuously monitored using a photodiode. Thephotodiode dark current was negligible. The output signalfrom the photodiode was amplified, digitized, and recorded.To record the hyperfine magnetic resonances, we swept Bcontinuously from one direction to the other and recorded thetransmitted light intensity. A Fabry-Pérot interferometer anda wavemeter were used to monitor the performance of thelaser.

The voltage output of the photodiode of Fig. 2 is V=V0 exp�−�Cs��op�A�l� where the parameter V0 is propor-tional to the laser light intensity and also includes scatteringand attenuation losses of the glass walls of the oven and theglass cell. The number density of Cs atoms, �Cs�, is mainlycontrolled by the temperature of the cell. The experimentalspecific absorption �A� is defined by l�A�=�0

l Tr��A�dz, av-eraged along all length elements dz of the optical paththrough the vapor. The density operator �=��z� can varyalong the optical path length through the vapor, since thelight can be attenuated substantially for �Cs��opl�1, espe-

PHYSICAL REVIEW A 75, 053415 �2007�

1050-2947/2007/75�5�/053415�6� ©2007 The American Physical Society053415-1

cially for low laser powers, and less so at higher powerswhere intense, circularly polarized light can burn throughthe vapor by pumping most of the atoms into the nonabsorb-ing end states. The specific absorption was determined fromfour measured photodiode voltages: �i� VC0, the voltage atthe photodiode produced by circularly polarized laserlight at room temperature when �Cs��op�A�l�1; �ii� VC

=VC0 exp�−�Cs��op�A�l�, the voltage at the photodiode pro-

duced by circularly polarized laser light at the temperature,magnetic field, microwave power, and polarization of theexperiment; �iii� VL0, the voltage at the photodiode producedby linearly polarized light at room temperature when�Cs��op�A�l�1; �iv� VL=VL0 exp�−�Cs��opl�, the voltage atthe photodiode produced by linearly polarized laser light atthe temperature of the experiment. The experimentallydetermined specific absorption was then given by

S1/2

P1/2σ +

-1/2 1/2(a)

D1

P3/2

(b)

S1/2

P1/2P3/2

3/2-3/2

σ + σ +

(m)

D2

1.00.80.60.40.2

2000150010005000

1.41.31.21.11.0

2000150010005000

1 3

(a.u.)

(a.u.)

σop

ν

ν

212

1 2 3

σop

AA

FIG. 1. �Color online� Energy sublevels of a hypothetical atom with I=0. Specific absorptions �1/2 A 1/2� and �−1/2 A −1/2� and theirrelative values are denoted by the diagonal arrows pointing up. Spontaneous decay and relative branching ratios are indicated by numbers onthe wavy arrows pointing down. Ground-state spin relaxations are denoted by the curved double-end arrows. The bars on each ground-statesublevel are steady-state occupation probabilities for �op=�s. The insets show that, as the optical pumping flux S increases, the specificabsorption �A� decreases monotonically from 1 to 0 for D1 pumping, and it increases monotonically from 1 to 3/2 for D2 pumping. Detuningthe laser from the maximum of the cross section to half maximum simply stretches out the curves.

iris

z

x

y

External cavitytunable diodelaser

ND filter

pellicle

λ2

λ4

photodiode

lens

x coily coil

z coil

ovencell

oscilloscopesynthesizer

oscilloscope

pellicle

wavemeter

Fabry-Perotinterferometer

horn

FIG. 2. �Color online� Apparatus �see text�.

GONG, JAU, AND HAPPER PHYSICAL REVIEW A 75, 053415 �2007�

053415-2

�A� = ln VC

VC0�� ln VL

VL0� . �3�

The high temperatures and the resulting high spin-exchangerates efficiently destroy the hyperfine polarizations �I ·S� andbirefringent polarizations �3fz

2− f�f +1�� that can be producedby linearly polarized pumping light �3�. This ensured thattaking �A�=1 for linearly polarized light was a good approxi-mation. In general cases where the spin-temperature distribu-tion is not necessarily valid, the denominator in Eq. �3�should be the value when the pumping intensity is lowenough not to polarize the atoms.

Figure 3 shows the observed specific absorption �A� ver-sus magnetic field B. Unlike the hypothetical atom with I=0 that was discussed in connection with Fig. 1, where �A��1 for any laser detuning, the data of Fig. 3 shows that forthe real Cs atom, with I=7/2, we observe �A��1 for a laserfrequency of 335 109.0 GHz �the lower trace�, which pumpspredominantly out of ground-state sublevels with total angu-lar momentum quantum numbers f = I+1/2=4, but for smalllaser powers we observe �A�1 for a laser frequency of335 122.0 GHz �the upper trace�, which pumps predomi-nantly out of ground-state sublevels with total angular mo-mentum quantum numbers f = I−1/2=3. At the 130 °C tem-perature of the cell, the unpolarized Cs vapor attenuated the335 109.0 GHz light by 2.9 e-foldings, and it attenuated the335 122.0 GHz light by 1.3 e-foldings. The circularly polar-ized incident laser power density S was 0.6–0.9 mW cm−2.In Fig. 3�a� the magnetic field of the microwaves oscillatesperpendicular to the static magnetic field and excites transi-tions with �m= ±1 and �f = ±1. In Fig. 3�b� the microwavefield oscillates nearly parallel to the static magnetic field, and

it can efficiently excite transitions with �m=0 and �f = ±1.The transitions are labeled with m, the average of the azi-muthal quantum numbers of the initial and final states. Thelarge resonance at zero magnetic field comes from the depo-larization induced by the transverse static magnetic field,which remains when the longitudinal field passes throughzero. This “zero dip” could be broader or narrower, depend-ing on how carefully we zeroed out the transverse field.

A second interesting conclusion that can be drawn fromFig. 3 is that the two sublevels 4m� and 3m� with the sameazimuthal spin quantum number m must have very nearly thesame populations, since the �m=0 resonances are not ob-served. The barely visible resonances that can be seen in thelower trace of Fig. 3�b� are actually resonances with �m= ±1 that result from a slight misalignment of the microwavehorn and the static magnetic field. Equal populations of thesublevels 4m� and 3m� are a characteristic of the spin-temperature distribution �4� when the sublevel populations,the diagonal elements of the atomic density operator �, are�fm � fm�=em /Z. Here =ln�1+ P�−ln�1− P� is the spin-temperature parameter, and P=tanh� /2�=2�Sz� is the elec-tron spin polarization. The spin partition function is Z= fmem. We expect the spin-temperature distribution to pre-vail when the spin exchange collision rate �se between pairsof alkali-metal atoms is larger than any other relaxation rateof the system, which was the case for our experiments, as wediscuss in more detail below.

Although the results of Fig. 3 were unexpected to us, theycan be understood quantitatively by noting that the densityoperator � for an optically pumped alkali metal obeys theevolution equation

72

52

32

72-

52-

32-

72-

52-

32-

72

52

32

72

52

32

72-

52-

32-

12

12-m

3210-1-2-3m

1.10

1.05

1.00

0.95

0.90

0.85

0.80

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

1.10

1.05

1.00

0.95

0.90

0.850.80

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8Magnetic field B (G)

(b)

335122.0 GHzσop

ν

335109.0 GHz ν

335122.0 GHz ν

335109.0 GHz ν

(a)

σop

σop

σop

AA

f=4

f=3

f=4

f=3

FIG. 3. �Color online� Specificabsorption �A� versus longitudinalmagnetic field B curves when theoscillating magnetic field is per-pendicular �a� and parallel �b� tothe static longitudinal field and thedirection of the laser beam. Uppertraces are for the laser frequencyof 335 109.0 GHz. Lower tracesare for 335 122.0 GHz. The twoinsets on the right show the cross-section curve for unpolarized at-oms and the frequency positionsof the two laser detunings. The in-sets on the left show the label mfor the possible transitions that areexcited by the oscillating mag-netic field.

MAGNETIC RESONANCE REVERSALS IN OPTICALLY… PHYSICAL REVIEW A 75, 053415 �2007�

053415-3

� =1

i��H,�� +

x

�x, �4�

where H=AgI ·S+ �gS�BSz−�IIz / I�Bz is the ground-stateHamiltonian for an alkali-metal atom in a magnetic field. Themagnetic dipole coupling coefficient is Ag, the nuclear spinoperator is I, the g value of the electron is very nearly gS=2.0023, the Bohr magneton is �B, and the magnetic dipolemoment of the nucleus is �I=1.579�N, where �N is thenuclear magneton. The contribution to the evolution from thespin relaxation mechanism x is denoted by �x, where themost important mechanisms are the following.

Optical pumping: the evolution �op due to optical pump-ing. This evolution mechanism is proportional to the opticalpumping rate �op of Eq. �1�. Without optical pumping, thesteady-state density matrix will be given by the Boltzmanndistribution �fm � fm�=e−Efm/kT /Z�T�, where Efm is the en-ergy of the sublevel fm�, T is the absolute temperature of thegas, k is Boltzmann’s constant, and the Boltzmann partitionfunction is Z�T�= fme−Efm/kT. The optical pumping efficiencyalso depends on how much of the polarization of the excited-state atoms returns to the ground state by spontaneous decayand quenching collisions with nitrogen molecules. Theseexcited-state processes are included in our computer calcula-tions.

Spin-exchange collisions: the evolution �se due to spin-exchange collisions between pairs of alkali-metal atoms atthe rate �se �5,6�. Spin-exchange collisions conserve the totalangular momentum of the vapor and drive the density matrixtoward the spin-temperature distribution mentioned above.�se is proportional to the number density of alkali-metal at-oms and therefore increases with temperature.

Spin-rotation collisions: the evolution �sr due to the spin-rotation interaction N ·S between alkali-metal atoms andbuffer-gas atoms or molecules �7,8�. This interaction couplesthe electron spin S to the rotational angular momentum N ofthe colliding pair and transfers spin to the translational an-gular momentum of the gas. The effects of the spin-rotationinteraction can be parametrized by the isotope-independentrate �sr �denoted by �SD in �8��.

Hyperfine-shift collisions: the evolution �hs due to the hy-perfine shift interaction �AI ·S that occurs during collisionsbetween an alkali-metal atom and a buffer-gas atom or mol-ecule �8–10�. This rate can be parametrized by the isotope-independent Carver rate �C. At relatively low magnetic fieldsof our experiments the hyperfine-shift collisions contributemostly to the widths of the microwave resonance lines �10�.Both �sr and �C are proportional to the number density of thebuffer gas, and �sr often depends strongly on the gas tem-perature.

Diffusion to the walls: the evolution �d due to the diffu-sion of the atoms to the cell walls, where most of the spinpolarization is lost. The effects of diffusion are often mod-eled approximately with a uniform damping rate �d=D / l2,where D is the diffusion coefficient and l is a characteristiclength, typically half the smallest dimension of the volumeof pumped atoms in the cell. For the experimental resultsshown in Fig. 3, we estimate that the spin-exchange rate was

�se=4.6�104 s−1. The other important spin relaxation ratesare the spin-rotation rate �sr=521 s−1, the Carver rate �C=107 s−1, the diffusion rate �d=47 s−1, and the opticalpumping rate �op=500 s−1.

As we will show below, one can obtain steady-state solu-tions to the evolution equation Eq. �4� with �=0 with com-puters, and the results are in very good agreement with ourobservations. But one can understand the reason for the cu-rious signal reversals without elaborate computer calcula-tions by considering the spin-temperature limit, when �se ismuch larger than any other characteristic rate of the system.Our experimental conditions for weak pumping light are ac-tually quite close to this limit. In the spin-temperature limit,the sublevel populations are determined by a single param-eter, which we will take to be the polarization P=tanh� /2�. Under these conditions, �A� will be a well-defined function of P, and one can readily see that �A�→1when P→0 and all atomic sublevels are equally populated,and �A�→0 when P→1 and all atoms are in the nonabsorb-ing sublevel 44�. How �A� changes from 1 to 0 with increas-ing P for various laser tunings can be understood with refer-ence to Fig. 4. Figures 4�a� and 4�b� show the two laserfrequencies, the specific absorptions �fm A fm� �narrowerbars�, and the sublevel occupation probabilities �fm � fm��wider bars�, for each ground-state sublevel. A spin-temperaturedistribution is assumed. In Fig. 4�a�, the laser is tunedto excite atoms predominantly from the upper hyperfinemultiplet with f =4. As the laser intensity increases, �A�= fm�fm A fm��fm � fm� decreases, because the popula-tions of weakly absorbing sublevels increase with increasinglaser intensity and increasing polarization. Figure 4�b� showsthe case where the laser is tuned to excite atoms predomi-nantly from the lower multiplet with f =3. If the pumpingintensity is not too great, the specific absorption �A� willincrease with increasing intensity, because the populations ofstrongly absorbing sublevels grow with the increasing polar-ization produced by the pumping laser. We already men-tioned that, irrespective of the laser detuning, sufficiently fastpumping with circularly polarized D1 light will cause thespecific absorption to vanish, �A�→0, because all atoms willbe pumped to the nonabsorbing “end state” 44�. So, if thelaser is tuned to cause maximum depopulation of atoms fromthe sublevels with f =3, the specific absorption will at firstincrease with increasing laser intensity S and increasing po-larization P, and will then decrease to zero.

In Fig. 4�c� we show the result of finding the steady-statesolutions of �4� with a computer program that takes the mostimportant relaxation processes mentioned above into ac-count. We also account for the spatial variation of the pump-ing rate due to attenuation of the laser beam in the vapor. Forthe laser detuning that pumps atoms predominantly fromsublevels with f =4, the specific absorption �A� decreasesmonotonically with increasing optical pumping rate. For thelaser frequency that pumps predominantly from sublevelswith f =3, �A� first increases with the pumping rate and thendecreases. The calculation �lines� agrees well with the ex-perimental data �solid circles in the figure�. Figure 4�d�shows the �A�-P curves for the two laser detunings. Here P is

GONG, JAU, AND HAPPER PHYSICAL REVIEW A 75, 053415 �2007�

053415-4

the average polarization, which is calculated using P=�op/ ��op+�sd+��d� and averaged along the beam path inthe vapor �along which �op decreases�. Here, � is defined as1+�= �Fz� / �Sz� �11�. The microwave resonance transitionswith �m= ±1 always remove spin from the vapor and causeP to decrease. If the laser is tuned to excite atoms predomi-nantly from the sublevels with f =4, when �A� decreasesmonotonically from 1 to zero with increasing P, we expectthe microwave resonances to always increase the specificabsorption for any laser power. The situation is more inter-esting when the laser is tuned to excite atoms predominantlyfrom the sublevels with f =3. Then if the laser intensity issmall enough that d�A� /dP�0, the microwave resonancesactually decrease the specific absorption and the vapor be-comes more transparent. But if the laser is intense enoughthat d�A� /dP�0, small microwave resonances correspond toan increase in the specific absorption.

Three points �A, B, and C� from the upper trace inFig. 4�d� are chosen to show the experimental results. Thedependence of specific absorption �A� on the magnetic fieldB is shown in Figs. 4�e�–4�g�. Point A is in the range whered�A� /dP�0 and �A��1. So both the magnetic resonancesand the zero dip show as absorption dips, as in Fig. 4�e�.

Moreover, since the atoms are weakly polarized and close tothe spin-temperature limit, the population difference betweenthe initial and final states of the resonant transition was aboutthe same for positive and negative azimuthal quantum num-bers, and magnetic resonances appeared on both sides of thezero dip. Point B is in the range where d�A� /dP�0 and�A��1. Since the microwave field causes only a smallchange in polarization, but the residual transverse field is bigenough to completely destroy the spin polarization, magneticresonances appear as positive absorption peaks and the zerodip appear as negative peaks, as is shown in Fig. 4�f�. PointC is in the range where d�A� /dP�0 and �A��1, so both thezero dip and magnetic resonances appear as absorptionpeaks, as in Fig. 4�g�. Because most of the atoms were instates with positive m for the more intense pumping light ofsituations B and C, we could barely observe magnetic reso-nances on the other side of the zero dip, since these corre-sponded to atoms with negative m.

Considering a more general case, the spin-temperaturedistribution does not always exist, we need to use detaileddensity-matrix modeling to calculate the resonance signals.For illustration, we tested a cell with a different buffer-gasmixture �50 torr N2 and 480 torr Ar�. The incident laserpower density S was about 0.1 mW cm−2 for all the laser

−4 −3 −2 −1 0 1 2 3 4m=

−3 −2 −1 0 1 2 3m=

f=3

f=4

−4 −3 −2 −1 0 1 2 3 4m=

−3 −2 −1 0 1 2 3m=

(a) (b)σop

ν ν

1.00.80.60.40.2

20x104151050Incident optical pumping rate (Hz)

(c)

Average polarization P

(d)A B

C

1.0

0.80.60.40.20.0

1.00.80.60.40.20.0

(e) (f)

Magnetic field B (G)

(g)1.000.950.900.850.80

0.80.40.0-0.4-0.8

1.051.041.031.021.011.00

0.80.40.0-0.4-0.8

1.081.061.041.021.00

0.80.40.0-0.4-0.8

A B C

σop

νσop

νσop

νσop

νσop

AA

A

FIG. 4. �Color online� �a�, �b��fm A fm� �narrower bar� and�fm � fm� �wider bar� for eachground-state sublevel at the twodifferent laser detunings. �c� Spe-cific absorption �A� versus opticalpumping rate curves. �d� Specificabsorption �A� versus average po-larization P curves. The uppertrace is for the laser frequency335 122.0 GHz, and the lowertrace is for the laser frequency335 109.0 GHz. Lines are calcula-tions and solid circles are experi-mental data. �e�, �f�, �g� �A�-Bcurves corresponding to points A,B, and C in �d�, respectively.

MAGNETIC RESONANCE REVERSALS IN OPTICALLY… PHYSICAL REVIEW A 75, 053415 �2007�

053415-5

detunings. The distance between the front and rear walls ofthe cell was 1.7 cm. The temperature was 90 °C. This lowertemperature reduces the number density of Cs atoms andleads to a much smaller spin-exchange rate. In spite of this,reversals of the magnetic resonance signals were also seen inthis cell. Figure 5 shows the comparison between the experi-mental data of Fig. 5�a� and the computer simulations of Fig.5�b� for four different laser frequencies. The simulated re-sults were calculated with the detailed model mentionedabove. The values of the modeling parameters were the op-tical pumping rate �op=20 s−1, the spin-exchange rate �se

=4500 s−1, the spin-rotation rate �sr=919 s−1, the Carver rate�C=107 s−1, and the diffusion rate�d=28 s−1. The simula-tion is strikingly similar to the experimental data.

In summary, we have experimentally demonstrated an un-usual signal reversal phenomena of the magnetic resonancesof ground-state alkali-metal atoms pumped by D1 circularlypolarized laser light. Under this circumstance, the hyperfinesplitting has to be at least partially optically resolved, and theoptical frequency of the pumping light should be tuned toreach near the lower hyperfine multiplets. The phenomenoncannot be explained by the conventional two- or three-levelmodelings, or even a multilevel modeling with a universaldamping rate. A valid theoretical explanation involves adensity-matrix modeling with two important spin-relaxationmechanisms, spin-rotation interaction and spin-exchange in-teraction. The reversal phenomena are more pronounced es-pecially when a rapid spin-exchange rate produces a spin-temperature distribution. These phenomena came to ourattention in connection with work on the Defense AdvancedResearch Projects Agency chip scale atomic clock program,where it is hoped that small, laser-pumped Cs or Rb vaporcells, similar to the ones of this experiment, could form thebasis of a useful new type of atomic clock. The relativelyhigh densities of alkali-metal vapors, which causes a highspin-exchange rate, and the narrow-line diode lasers used topump the miniature cells of CSAC clocks often lead to con-ditions similar to those reported here. For microwave-interrogation atomic clocks, some care should be exercisednot to operate these systems with parameters close to thosethat reduce or reverse the resonance signals.

This work was supported by the Air Force Office of Sci-entific Research, with supplemental support from the De-fense Advanced Projects Agency.

�1� W. Happer, Rev. Mod. Phys. 44, 169 �1972�.�2� J. Fricke, J. Haas, E. Lüshcer, and F. A. Franz, Phys. Rev. 163,

45 �1967�.�3� B. S. Mathur, H. Tang, and W. Happer, Phys. Rev. A 2, 648

�1970�.�4� L. W. Anderson and A. T. Ramsey, Phys. Rev. 124, 1862

�1961�.�5� L. C. Balling, F. M. Pipkin, and R. J. Hanson, Phys. Rev. 133,

A607 �1964�.�6� F. Grossetête, J. Phys. �Paris� 25, 383 �1964�.

�7� R. A. Bernheim, J. Chem. Phys. 36, 135 �1962�.�8� D. K. Walter, W. M. Griffith, and W. Happer, Phys. Rev. Lett.

88, 093004 �2002�.�9� J. Vanier and C. Audoin, The Quantum Physics of Atomic Fre-

quency Standards �A. Hilger, Philadelphia, 1989�.�10� P. J. Oreto, Y.-Y. Jau, A. B. Post, N. N. Kuzma, and W. Hap-

per, Phys. Rev. A 69, 042716 �2004�.�11� S. Appelt, A. B. Baranga, C. J. Erickson, M. Romalis, A. R.

Young, and W. Happer, Phys. Rev. A 58, 1412 �1998�.

1.02

1.01

1.00

0.99

0.98

0.97-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

335119.5 GHz

335116.8 GHz

335115.0 GHz

335112.3 GHz

1.02

1.01

1.00

0.99

0.98

0.97-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

Magnetic field B (G)

(a)

(b)

AA

FIG. 5. �Color online� Experimental results and simulations fora different cell with buffer-gas mixture of 50 torr N2 and 480 torrAr. �a� Measured �A� versus magnetic field B curves at four differ-ent laser frequencies. �b� Simulation results.

GONG, JAU, AND HAPPER PHYSICAL REVIEW A 75, 053415 �2007�

053415-6